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Matrix Operations
Chapter 7
Matrix Mathematics
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Matrix Mathematics
• Matrices are very useful in engineering
calculations. For example, matrices are used to:
– Efficiently store a large number of values (as we have
done with arrays in MATLAB)
– Solve systems of linear simultaneous equations
– Transform quantities from one coordinate system to
another
• Several mathematical operations involving
matrices are important
Engineering Computation: An Introduction Using MATLAB and Excel
Review: Properties of Matrices
• A matrix is a one-or two dimensional array
• A quantity is usually designated as a matrix by
bold face type: A
• The elements of a matrix are shown in square
brackets:
Engineering Computation: An Introduction Using MATLAB and Excel
Review: Properties of Matrices
• The dimension (size) of a matrix is defined by the
number of rows and number of columns
• Examples:
3 × 3:
2×4:
Engineering Computation: An Introduction Using MATLAB and Excel
Review: Properties of Matrices
• An element of a matrix is usually written in lower
case, with its row number and column number as
subscripts:
• In MATLAB, an element is designated by the
matrix name with the row and column numbers in
parentheses: A(1,2)
Engineering Computation: An Introduction Using MATLAB and Excel
Matrix Operations
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Matrix Addition
Multiplication of a Matrix by a Scalar
Matrix Multiplication
Matrix Transposition
Finding the Determinant of a Matrix
Matrix Inversion
Engineering Computation: An Introduction Using MATLAB and Excel
Matrix Addition
• Vectors must be the same size in order to add
• To add two vectors, add the individual elements:
• Matrix addition is commutative:
A+B=B+A
Engineering Computation: An Introduction Using MATLAB and Excel
Multiplication of a Matrix by a Scalar
• To multiple a matrix by a scalar, multiply each
element by the scalar:
• We often use this fact to simplify the display of
matrices with very large (or very small) values:
Engineering Computation: An Introduction Using MATLAB and Excel
Multiplication of Matrices
• To multiple two matrices together, the matrices
must have compatible sizes:
This multiplication is possible only if the number
of columns in A is the same as the number of rows
in B
• The resultant matrix C will have the same number
of rows as A and the same number of columns
as B
Engineering Computation: An Introduction Using MATLAB and Excel
Multiplication of Matrices
• Consider these matrices:
• Can we find this product?
Yes, 3 columns of A = 3 rows of B
• What will be the size of C?
2 X 2: 2 rows in A, 2 columns in B
Engineering Computation: An Introduction Using MATLAB and Excel
Multiplication of Matrices
• Easy way to remember rules for multiplication:
These values
must match
Size of Product
Matrix
Engineering Computation: An Introduction Using MATLAB and Excel
Multiplication of Matrices
• Element ij of the product matrix is computed by
multiplying each element of row i of the first
matrix by the corresponding element of column j
of the second matrix, and summing the results
• This is best illustrated by example
Engineering Computation: An Introduction Using MATLAB and Excel
Example – Matrix Multiplication
• Find
• We know that matrix C will be 2 × 2
• Element c11 is found by multiplying terms of row 1
of A and column 1 of B:
Engineering Computation: An Introduction Using MATLAB and Excel
Example – Matrix Multiplication
• Element c12 is found by multiplying terms of row 1
of A and column 2 of B:
Engineering Computation: An Introduction Using MATLAB and Excel
Example – Matrix Multiplication
• Element c21 is found by multiplying terms of row 2
of A and column 1 of B:
Engineering Computation: An Introduction Using MATLAB and Excel
Example – Matrix Multiplication
• Element c22 is found by multiplying terms of row 2
of A and column 2 of B:
Engineering Computation: An Introduction Using MATLAB and Excel
Example – Matrix Multiplication
• Solution:
Engineering Computation: An Introduction Using MATLAB and Excel
Practice Problems
• Find C = AB
Engineering Computation: An Introduction Using MATLAB and Excel
Practice Problems
• Find C = AB
Engineering Computation: An Introduction Using MATLAB and Excel
Practice Problems
• Find C = AB
Engineering Computation: An Introduction Using MATLAB and Excel
Matrix Multiplication
• In general, matrix multiplication is not
commutative:
AB ≠ BA
Engineering Computation: An Introduction Using MATLAB and Excel
Transpose of a Matrix
• The transpose of a matrix by switching its row and
columns
• The transpose of a matrix is designated by a
superscript T:
• The transpose can also be designated with a prime
symbol (A’). This is the nomenclature used in
MATLAB
Engineering Computation: An Introduction Using MATLAB and Excel
Determinant of a Matrix
• The determinant of a square matrix is a scalar
quantity that has some uses in matrix algebra.
Finding the determinant of 2 × 2 and 3 × 3
matrices can be done relatively easily:
• The determinant is designated as |A| or det(A)
• 2 × 2:
Engineering Computation: An Introduction Using MATLAB and Excel
Determinant of a Matrix
• Examples:
Engineering Computation: An Introduction Using MATLAB and Excel
Determinant of a Matrix
• 3 × 3:
• Similar for larger matrices, but easier to do with
MATLAB or Excel
Engineering Computation: An Introduction Using MATLAB and Excel
Inverse of a Matrix
• Some square matrices have an inverse
• If the inverse of a matrix exists (designated by -1
superscript), then
where I is the identity matrix – a square matrix with
1’s as the diagonal elements and 0’s as the other
elements
Engineering Computation: An Introduction Using MATLAB and Excel
Inverse of a Matrix
• The inverse of a 2X2 matrix is easy to find:
Engineering Computation: An Introduction Using MATLAB and Excel
Inverse of a Matrix
• Example: find inverse of A:
Engineering Computation: An Introduction Using MATLAB and Excel
Check Result
Engineering Computation: An Introduction Using MATLAB and Excel
Practice Problem
• Find A-1, check that A A-1 = I
Engineering Computation: An Introduction Using MATLAB and Excel
Inverse of a Matrix
• Note from the formula for the inverse of a 2 × 2
matrix that if the determinant equals zero, then
the inverse is undefined
• This is true generally: the inverse of a square
matrix exists only of the determinant is non-zero
Engineering Computation: An Introduction Using MATLAB and Excel